(This topic is also in Section 1.4 in Finite Mathematics, Applied Calculus and Finite Mathematics and Applied Calculus)
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This is a continuation of a tutorial on linear models. To go back to the start of the tutorial, press here.
Using linear functions to describe or approximate relationships in the real world is called linear modeling. Here, we study several kinds of linear model:
It is often observed that, the more you charge for an item, the lower the demand will be: As the price goes up, the demand goes down.
Q How do we measure demand?
A A common way of measuring demand for an item is by the number of items you sell.in some period of time. For instance, you can measure the demand for Starship Troopers T shirts by daily sales.
Linear Demand Function
A demand equation or demand function expresses demand q (the number of items demanded) as a function of the unit price p (the price per item). A linear demand function has the form q = mp + b
Interpretation of m
Interpretation of b
Example If the demand for T-shirts, measured in daily sales, is given by q =-4p + 90,where p is the sale price in dollars, then daily sales drop by four T-shirts for every $1 increase in price. It the T-shirts were given away, the demand would be 90 T-shirts per day. |
A demand function gives the number of items consumers are willing to buy at a given price, and a higher price generally results in a lower demand. However, as the price rises, suppliers will be more inclined to produce these items (as opposed to spending their time and money on other products), so supply will generally rise. A supply function gives q, the number of items suppliers are willing to make available for sale, as a function of p, the price per item.
Linear Supply Function, Equilibrium Price
A supply equation or supply function expresses q (the number of items suppliers are willing to make available) as a function of the unit price p (the price per item). A linear demand function has the form q = mp + b
Interpretation of m
Interpretation of b
Example The number of T-shirts I am prepared to tie-dye and supply to Campus Creations Inc. per day depends on the price, $p, I obtain according to q = 2.5p + 5.For every $2 increase in price, I am willing to supply 5 additional shirts per day.
Equilibrium Price
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Q What happens if the price is set lower or higher than the equilibrium price?
A To illustrate what happens at various prices, let us look at the graphs of demand and supply for Luddington's Wellington Boots.
Things all around us change with time. Thus, it is natural to think of many quantities, such as your income or the temperature in Honolulu, as functions of time. We usually use the independent variable t to denote time (in seconds, hours, days, years, etc.). If a quantity q changes with time, then q is a function of t. Here, we are interested in linear functions of t:
q = mt + b q is a linear function of time tLet's go through such an example.
In view of the above, we interpret the slope as the rate of change of the quantity q with respect to time t. Here is a summary of what we saw above, together with something new.
Linear Change Over Time
If a quantity q is a linear function of time t, so that q(t) = mt + b,then the slope m measures the rate of change of q, and b is the quantity at time t = 0, the initial quantity. If q represents the position of a moving object, then the rate of change is also called the velocity.
Units of m
Examples
2. You are driving down the Ohio Turnpike such that the number of miles you have traveled after t hours is given by s(t) = 54t + 20. Then your speed is 54 miles per hour, and at time t= 0 you had traveled 20 miles. |
All of the preceding examples share the following common theme.
General Linear Models
If y = mx + b is a linear model of changing quantities x and y, then the slope m is the rate at which y is increasing per unit increase in x, while the y-intercept b is the value of y that corresponds to x = 0. The slope m is measured in units of y per unit of x, while the intercept b is measured in units of y. |
Now try some of the exercises in Section 1.4 of the textbook, or press "Review Exercises" on the sidebar to see a collection of exercises that covers the whole of Chapter 1.